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| Mirrors > Home > MPE Home > Th. List > al0ssb | Structured version Visualization version GIF version | ||
| Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.) |
| Ref | Expression |
|---|---|
| al0ssb | ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 2 | sseq2 4010 | . . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅)) | |
| 3 | ss0b 4401 | . . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅) | |
| 4 | 2, 3 | bitrdi 287 | . . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)) |
| 5 | 1, 4 | spcv 3605 | . 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅) |
| 6 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ 𝑦 | |
| 7 | 6 | ax-gen 1795 | . . 3 ⊢ ∀𝑦∅ ⊆ 𝑦 |
| 8 | sseq1 4009 | . . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦)) | |
| 9 | 8 | albidv 1920 | . . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦)) |
| 10 | 7, 9 | mpbiri 258 | . 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦) |
| 11 | 5, 10 | impbii 209 | 1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 ⊆ wss 3951 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 |
| This theorem is referenced by: iota0def 47050 aiota0def 47108 |
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